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100 V NORMED SPACES
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that H n Bno 7^ 0, and let n be an arbitrary integer ^ nQ . For each k < n,
let Jk be a finite subset of B* containing H n Bfc, and such that for any finite subset Lk of Bjt containing Jfc5 we have \\zk- £*J < e/(« + 1) (5.3.4). |
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Then, if L = M L*, we have £ z* ~ £ *J ^ e» anc* as L r) H, it follows
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aeL
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from the definition of H that £ zk — ]T xa < 3e, which ends the proof.
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There is a similar (and easier) result when A is decomposed in a finite
number of subsets Ek (I ^ k ^ n); moreover, in that case, there is a converse to (5.3.6), namely, if each of the families COa6Bfc *s absolutely summable, so is COaeAJ ^e proof follows, by induction on n, from the criterion (5.3.4). |
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PROBLEMS
1. Let (dn) be a sequence of real numbers dn ^ 0, such that the series (dn) is not convergent
R
(i.e., Km J) dk = +00). What can be said of the convergence of the following series:
d. da d, da _ |
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2. Let (wtt) be a convergent series of real numbers, which is not absolutely convergent, and
00
let s = 2 un . For each number s' ^ s, show that there exists a bijection o* of N onto itself
n = 0
oo
such that a(n) = n for all n such that un ^ 0, and that X wff(fl) = 5'. (Show by induction
n = 0
that for each n there is a bijection an of N onto itself such that an(k) — k for all k such
that wfc ^ 0 and that, if i/in) = wffn(*) , there is an index pn having the property that, for k ^ pn |
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furthermore, orn+i is such that an+i(/c) = an(k) for all /: such that an(k)<pn and all &
such that I/* ^ — 1/«.)
3. Show that for every finite family (xj)i6l of points of the product space R" (with the
norm IWI = sup|&| for x= (^)i^fc<n), one has T ||xt|| ^2/i • sup ||T^i|| (consider
lei Jcl feJ
first the case n — 1).
4, In a normed space E, a series (*„) is said to be commutatively convergent if, for every
bijection a of N onto itself, the series (x0^ is convergent.
(a) In order that a convergent series (*„) be commutatively convergent, it is necessary
and sufficient that for every e > 0, there exist a finite subset J of N such that, for any subset H of N for which J n H = 0, || £ jcn || ^ e. When that condition is satisfied, the
.00 ^ •
sum 2 x"(n) is independent of CT. (To prove the last assertion, and the sufficiency of the
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